SUMMARY

Cormorants are water birds that forage by submerged swimming in search and
pursuit of fish. Underwater they swim by paddling with both feet
simultaneously in a gait that includes long glides between consecutive
strokes. At shallow swimming depths the birds are highly buoyant as a
consequence of their aerial lifestyle. To counter this buoyancy cormorants
swim underwater with their body at an angle to the swimming direction. This
mechanical solution for foraging at shallow depth is expected to increase the
cost of swimming by increasing the drag of the birds. We used kinematic
analysis of video sequences of cormorants swimming underwater at shallow depth
in a controlled research setup to analyze the swimming gait and estimate the
resultant drag of the birds during the entire paddling cycle. The gliding drag
of the birds was estimated from swimming speed deceleration during the glide
stage while the drag during active paddling was estimated using a mathematical
`burst-and-glide' model. The model was originally developed to estimate the
energetic saving from combining glides with burst swimming and we used this
fact to test whether the paddling gait of cormorants has similar
advantages.

We found that swimming speed was correlated with paddling frequency
(r=0.56, P<0.001, N=95) where the increase in
paddling frequency was achieved mainly by shortening the glide stage
(r=–0.86, P<0.001, N=95). The drag
coefficient of the birds during paddling was higher on average by two- to
threefold than during gliding. However, the magnitude of the drag coefficient
during the glide was positively correlated with the tilt of the body
(r=0.5, P<0.003, N=35) and negatively correlated
with swimming speed (r=–0.65, P<0.001,
N=35), while the drag coefficient during the stroke was not
correlated with tilt of the body (r=–0.11, P>0.5,
N=35) and was positively correlated with swimming speed
(r=0.41, P<0.015, N=35). Therefore, the
difference between the drag coefficient during the glide and during propulsion
diminished at lower speeds and larger tilt. The mean drag of the birds for a
single paddling cycle at an average swimming speed of 1.5 m
s–1 was 5.5±0.68 N. The burst-and-glide model predicts
that energy saving from using burst-and-glide in the paddling cycle is limited
to relatively fast swimming speeds (>1.5 m s–1), but that
as the birds dive deeper (>1 m where buoyancy is reduced), the
burst-and-glide gait may become beneficial even at lower speeds.

Cd has been measured for different animals from models
and carcasses in a water or air tunnel
(Fish, 1984; Hui, 1988a;
Lovvorn et al., 2001;
Maybury and Rayner, 2001) or
by using trained animals towed behind a boat
(Williams and Kooyman, 1985).
An indirect approach is to measure drag from the deceleration of live animals
as they glide (Bilo and Nachtigall,
1980; Clark and Bemis,
1979; Miller et al.,
2004; Stelle et al.,
2000; Williams and Kooyman,
1985). This technique uses the fact that during the glide no
thrust is being produced, hence the deceleration results directly from drag.
By measuring the deceleration of the body, the drag force can be calculated
from Newton's second law (F=Ma, where F=force,
M=mass and a=acceleration). However, some correction is
required for the added mass of water that is accelerated with the body
(Vogel, 1994). The advantage
of this technique is that the estimate is derived from animals swimming
voluntarily, cruising at preferred speeds so that the shape and orientation of
body parts is realistic. However, unlike with rigid objects,
Cd of live animals changes during swimming. The swimming
motions (undulations, flapping, rowing and paddling) increase the drag
significantly beyond that of a motionless animal at the same swimming speed
(Weihs, 1974;
Schultz and Webb, 2002;
Vogel, 1994). None of the
above-mentioned techniques considers the elevated drag during active swimming
and the drag measured is thus termed `parasite drag' (for models and carcasses
where the propelling appendages are removed) or `passive drag' (for gliding
animals). A different approach that does allow the estimation of the drag for
active swimming is to model the thrust from the kinematics of the propelling
appendages. Once thrust is estimated, then during swimming at constant speed,
drag should equal thrust (Fish,
1993; Vogel,
1994). However, since estimation of passive drag from glides is
relatively simple, it would be beneficial if active drag could be estimated
directly from passive drag using simple kinematics. We explore this option
using the swimming behavior termed `burst-and-glide' (described by
Weihs, 1974) that has been
shown to exist in fish, marine mammals, penguins, water birds, lobsters,
plankton and numerous aerial flyers (Haury
and Weihs, 1976; Hove et al.,
2001; Lovvorn,
2001; Sato et al.,
2003; Spanier et al.,
1991; Van Dam et al.,
2002; Videler and Weihs,
1982; Webb and Fairchild,
2001; Weihs, 2002,
2004).

Swimming animals that use burst-and-glide can reduce the total energetic
cost of swimming (Weihs, 1974;
see Appendix). With this strategy, instead of swimming continuously at a
constant speed, the swimmer accelerates to a final speed
(Uf) above the desired average speed and then glides
(decelerates) until reaching the initial speed (Ui). The
energetic saving from such behavior is calculated as the ratio (R)
between the energy required to burst-and-glide and the energy required to swim
at the same average speed but at constant speed. R can be calculated
directly from the burst-and-glide model using easily obtained kinematic data
(see, for example, Videler and Weihs,
1982). However, the model requires knowledge of the parameterα
, which is the ratio of drag during active swimming and drag during
gliding. For example, the α used by Weihs
(1974) for fish was 3, based
on evidence that the drag of active swimming fish (dace and trout) is about
three times the drag during passive gliding
(Lighthill, 1971;
Webb, 1971). A similar value
of α was reported later for dolphins
(Fish, 1993).

Cormorants propel their body underwater by stroking (paddling) with both
feet simultaneously while the wings are tightly folded against the body
(Lovvorn, 1991;
Schmid et al., 1995). However,
when swimming on the surface, cormorants employ alternate paddling, stroking
with one foot at a time (Lovvorn,
1991; Ancel et al.,
2000). Kinematic data on cormorants (Phalacrocorax carbo
sinensis Blumelbach) swimming underwater
(Ribak et al., 2004) showed
that the paddling cycle is divided into three distinct successive stages: a
stroke stage, where both feet move backward and generate the thrust; a glide
stage, where the feet have no motion relative to the body; and a recovery
stage, where the feet move forward relative to the body, in preparation for
the next stroke. The glide stage lasts for approximately 50% of the entire
paddling cycle duration (Ribak et al.,
2004). The combination of a short power stage, where the birds
accelerate, with a long glide stage, where the birds decelerate, resembles the
burst-and-glide swimming pattern described above.

Submerged cormorants are positively buoyant
(Lovvorn and Jones, 1991) so
that the birds need to invest energy to remain submerged. Most of the buoyancy
of water birds is due to air volumes in the plumage and in the airsacs
(Lovvorn and Jones, 1991;
Wilson et al., 1992), and
their buoyancy therefore depends on diving depth. As the birds dive deeper,
the hydrostatic pressure compresses the gas volumes, making the birds less
buoyant. During shallow horizontal dives, cormorants swim with the midline of
the body at an angle to the swimming direction (pitch, head pointing down,
tail pointing up). This angle-of-attack generates hydrodynamic lift to help
offset the positive buoyancy of the birds. This tilt of the body is highest
during the stroke and recovery stages
(Ribak et al., 2004). Using
lift to counter buoyancy by tilting the body allows the birds to forage
underwater at shallow depths where their buoyancy is maximal, but this
behaviour should result in elevated drag during submerged swimming.

The drag coefficient (Cd) of the Brandt's cormorant
Phalacrocorax pencillatus was measured by Lovvorn et al.
(2001) by towing frozen, dead
birds and a smooth featherless model. The values they obtained (based on the
surface area as the characteristic area, A) were about 0.008 and
0.022 for the model and carcasses, respectively (at
Re=106; figs 6 and 7 in
Lovvorn et al., 2001).
However, these values are applicable only for gliding birds at depths where
buoyancy is negligible (≫20 m; Wilson
et al., 1992), since the towed birds were aligned in the direction
of the flow and were not performing swimming motions. Great cormorants are
shallow divers, mostly diving to depths <11 m
(Grèmillet et al., 1999;
Johansgard, 1993). Therefore,
a method to estimate the drag during active paddling while taking into account
the influence of buoyancy at shallow depths will provide a more realistic
evaluation of the mechanical constraints on swimming for these waterbirds.

The problem of increased drag resulting from propulsive motions is a
general issue confronting all animals that move in fluid media (air, water,
biological fluids) and different species have found various ways of coping.
Here we examine the adaptations of the cormorant, a flyer and also an adept
swimmer. We use the long glide stage between the stroke and recovery stages in
the paddling cycle of the cormorant to measure the passive drag of the birds
from their deceleration. Then, we employ the `burst-and-glide' model to
calculate the drag of the bird during active swimming and to evaluate the
energetic benefit of adding glides to the paddling cycle.

Materials and methods

Morphometric measurements

The calculation of Cd from deceleration requires a
typical area (A, in Eqn 1) and added mass coefficients for the body
of great cormorants Phalacrocorax carbo sinensis Blumelbach. We used
volume V to the 2/3 power (V2/3) for area, by
calculating the volume of each bird from its body mass, and from the average
density (810 kg m–3) reported for cormorant carcasses
(Ribak et al., 2004). An
earlier study on parasite drag of cormorants reported drag coefficients based
on surface area (Lovvorn et al.,
2001), so we measured this area as well, to allow comparison (see
below). Five carcasses of adult cormorants that were shot over fishing ponds
during a wildlife management program were obtained fresh and stored frozen
(–20°C) until processing after completely defrosting at room
temperature (∼15°C). The carcasses were used to estimate the added
mass, surface area and general shape of the birds in three dimensions. This
was done by folding the wings to their normal position during swimming, and
securing them in place with tape. Each carcass was laid with its neck
stretched on a 0.5 cm thick, 1×0.1 m plate, with markings every 1 cm
along the long side of the plate. The tip of the bill, the tip of the tail and
the center of the body were secured to the plate with tape. The plate with the
bird was then advanced, 1 cm at a time, through a measuring device consisting
of two perpendicular metal rods, sliding orthogonally with respect to each
other (Fig. 1A). Each rod was
scaled using a measuring tape (±1 mm), allowing the simultaneous
measurement of height and width of the carcass each centimeter along the long
axis of the body. The body is covered with soft plumage and has some
flexibility, thus the measured height and width may change with the pressure
applied by the measuring tool. To minimize this error, prior to each
measurement the rods were pressed lightly onto the carcass and then released
gently to allow the plumage and body tissues to push the rods back to normal
position. The set of measurements from each carcass thus included diameters of
1 cm thick transverse `slices' of the birds
(Fig. 1B). The volume of each
slice was calculated from the area of both sides of the slice, using the area
of an ellipse (area = π yz, where y is half the height
measurement and z is half the width measurement). Then, the area of
both sides was averaged and multiplied by the thickness of the slice (1 cm) to
obtain volume. The volumes of all the slices were summed to yield the total
volume of the carcass. Similarly, surface area was measured from the
perimeters of each side of the slices using the equation for an ellipse:
(2)
The surface area of each slice was assumed to be the average of both
perimeters, multiplied by the slice thickness (1 cm). Then, all areas along
the length of the bird were summed to yield the surface area of the whole
bird. We divided the quantity V2/3 of the carcasses by the
measured surface area and used this ratio to convert Cd
calculated based on V2/3 to Cd
calculated based on surface area in the live birds.

Shape of the cormorant's body in three dimensions. (A) Schematic
illustration (not to scale) of the apparatus used to measure the diameters of
the cormorant carcass along the body's main axis.(B) Diameter of the width
(lateral axis, solid rectangles) and height (dorso–ventral axis, empty
circles). The x-axis represents the position along the body's main
axis in % of body length (total body length = 85 cm). The diameters were used
to calculate surface area, volume and added mass coefficients. The broken
vertical lines mark the definition of the body excluding the neck and tail
(trunk).

The added mass is the mass of water that is accelerated with a body and is
proportional to the volume of the body
(Denny, 1993). The exact
magnitude of the Cd calculated from decelerations depends
slightly on the estimation of the added mass. Underestimation of the added
mass will result in a lower calculated value of the drag coefficient and
vice versa. To be as accurate as possible we estimated the added mass
directly from the size and shape of the birds and accounted for the
angle-of-attack θ of the body and tail. The added mass coefficient is
the volume of the added mass divided by the volume of the body. An infinitely
long cylinder moving parallel to its long axis has an added mass coefficient
close to zero while the same cylinder moving perpendicular to its length will
have an added mass coefficient of 1 (Lamb,
1945). Similarly the added mass of the body of the cormorant
depends on whether it moves aligned in the direction of swimming (parallel to
the length of the body) or if the body's main axis is tilted at an angle to
the flow. Since the birds in our experiment were always gliding with their
body at an angle-of-attack to the flow (tail pointing up) we obtained two
added mass coefficients: once for the cormorant moving lengthwise and a second
time for motion perpendicular to the longitudinal axis of the bird
(θ=90°). The values of both were combined to estimate the added mass
coefficient for a specific angle-of-attack. For the sake of this analysis we
disregarded the neck, which contributes less than 10% to the volume of the
entire body (9.5±1%, N=5), does not orient as the rest of the
body, and therefore resembles a cylinder aligned with the direction of flow.
We thus focused on estimating the added mass of the body (trunk) and tail. For
the added mass coefficient relevant for lengthwise motion
(Cax) of the body we used 0.122, which is the theoretical
value for a prolate spheroid with fineness ratio (length/diameter) of 3 (p.
155 in Lamb, 1945). For motion
perpendicular to the flow (downwards) we used Cay=1.339
(Table 1) for the body, which
was the average calculated from our carcasses [standard error of mean
(s.e.m.)=0.045, N=5] using elongated
body theory (see, for example, Lighthill,
1970,
1960). The theory states that
the added mass of each cross section of a slender body moving broadside to the
flow is approximately the volume of a cylinder with a circular cross section
that encompasses the diameter of the body at the specific cross section. Thus
using the diameters of the body measured at each 1 cm along its length
(excluding the neck and tail), we calculated the volume of encompassing
cylinders (1 cm long) and integrated all volumes to achieve the volume of
added mass when the body is tilted at θ=90°.

The volume of added mass was then divided by the actual volume of the bird
to yield the added mass coefficient for motion perpendicular to the flow
(Cay). The specific added mass coefficient for a givenθ
was assumed to be the sum of the projections of the two added mass
coefficients (Cax and Cay) on the
direction of swimming:
(3)
The same approach was used for the tail, treating it as a flat plate with
trapezoid shape and dimensions of 2 cm and 8.6 cm at the bases and 13.8 cm as
the length (data taken from Ribak et al.,
2004, which described the planform area of tails of cormorants
during swimming). For such a plate Cax is negligible
(Cax=0) and Cay is the volume of a
truncated cone with radii 1 cm and 4.3 cm at the bases and height 13.8 cm
(=3.44×10–4 m3) divided by the volume of the
bird. The added mass coefficient Ca(θ), for the tail
is calculated as in Eqn 3 but using the angle-of-attack of the tail (the angle
between the upper surface of the tail and the direction of swimming).
Calculating separate added mass coefficients for the body and tail enabled us
to account for changes in the added mass of the bird due to changes in the
angle-of-attack of the body and of the tail when the two were not the same.
Finally the added mass coefficient Ca of the entire bird
(body and tail) is the sum of the coefficients
(Ca(θ)) for the tail and for the body.

Body length (tip of bill to tip of tail) of the live birds used to
calculate the Reynolds number was measured from scaled images of the birds
swimming with their neck stretched.

Swimming sequences

Video sequences were obtained from 5 male and 5 female adult (age >3
years) great cormorants swimming in an experimental setup described previously
(Ribak et al., 2004). The
birds were filmed swimming inside a straight, 7 m-long metal mesh channel,
with a rectangular 0.5 m×0.5 m profile, placed on the floor of a 1
m-deep pool. A 2 m-long section in the tunnel, starting 3.5 m away from the
entrance, was used as a testing section. It was equipped with a mirror (2
m×0.7 m) angled at 45° above the tunnel. The birds were filmed
swimming in the test section using a CCTV video camera (VK-C77E, Hitachi,
Tokyo, Japan; 752×582 effective pixels, shutter speed 1/50s), inside an
underwater housing, and connected to a S-VHS video (HR-S7600AM, JVC, Tokyo,
Japan). The camera was positioned 0.5 m above bottom, 2 m away from the middle
of the testing section, and covered most of the testing section and the mirror
(2.3 mm/pixel in the side view 2.7 mm/pixel in the upper view,
0.0030–0.0036 body length/pixel, respectively), thus allowing
simultaneous lateral and dorsal (through the mirror) views of the swimming
bird while using a single camera. The video sequences (each containing one
complete paddling cycle) were converted to consecutive video fields (50 fields
s–1, i.e. time resolution was ±0.02 s) using a video
editing system (Edit 6, Autodesk Inc., San Rafael, CA, USA).

The position of the test section along the tunnel was chosen to allow
sufficient distance for the birds to develop a repeated burst-and-glide cycle
at given average swimming speed in a straight path prior to the measurement of
swimming parameters. The tunnel was positioned along the floor at the center
of the pool, at least 1 m away from the nearest wall. We calculated the ground
effect (Hoerner, 1975) on the
cormorants and found it to be negligible beyond a distance of one body
thickness (maximum distance between dorsal and ventral sides of the body) from
the floor (at Re>106) and hence only used runs that
were >10 cm distant from the floor. The upper 0.5 m height limitation of
the tunnel ensured that the birds were swimming away from the surface, at a
distance of more than 4 times the body thickness. Hence, no surface waves
formed by the swimming bird were measured and the resulting added drag can be
assumed to be negligible (approaching zero at depth >3 times the body
thickness; fig. 289 in Hertel,
1966).

Only a few minutes each time (<10 min) were required to obtain 5–8
video sequences from each bird in a series of 4 sampling days. The short
duration of the experiments ensured that changes in buoyancy due to
time-dependant plumage wettability would be small
(Ribak et al., 2005).

Kinematic analysis

In each sequence, stride length was defined as the horizontal distance
traversed during the paddling cycle. The cycle duration was calculated from
the number of video fields per cycle and paddling frequency was defined as
(cycle duration)–1. The average swimming speed for the entire
cycle was defined as stride length divided by cycle duration. Each paddling
cycle was divided into three distinct stages: stroke, glide and recovery,
based on the motion of the feet as described above and in Ribak et al.
(2004). The changes in gait
during swimming were analyzed by testing for correlation between the duration
of each of the three stages with paddling frequency.

The upper view (through the mirror) in each video field was used to measure
the instantaneous speed of the birds. This was done by measuring in each video
field the position of a small, round (1.5 cm in diameter) tag glued to the
feathers on the back of the birds. The momentary horizontal swimming speed was
calculated as the first derivative (∂x/∂t) of the
function describing the change in position along the x (horizontal)
axis with time, using a Lagrange 3-points numerical equation
(Hildebrand, 1956). In each
paddling cycle, the birds had slight vertical deviations (∼2 cm) from the
horizontal direction (Ribak et al.,
2004). However, the descent and ascent angles (measured from the
side view) were sufficiently small (∼4°) that the horizontal component
of the measured speed represents over 99% of the two-dimensional speed. The
angle-of-attack of the body during the glide was defined as the angle between
the longitudinal midline of the body (base of neck to base of tail) and the
instantaneous direction of swimming in the side view.

The velocity profiles, each comprising one complete paddling cycle
(Fig. 2), were used for
analysis of drag and burst-and-glide. During the stroke, the body accelerates,
and during the glide and recovery it decelerates. Following Weihs
(1974), we defined a parameter
Uf as the maximum instantaneous speed of the body during
the stroke, and Ui as the minimum speed at the start of
the stroke. The time interval between them was defined as the power stage
(tp, see Fig.
2) to differentiate it from the stroke stage, which is a more
robust kinematic definition based solely on the motion of the feet. The speed
at the start of the recovery stage was defined as Ur and
tg was defined as the duration of the deceleration from
Uf to Ur. tr was
the duration of the recovery stage. We measured Lp,
Lg and Lr as the horizontal distances
traversed during the power (tp), glide
(tg) and recovery (tr) stages,
respectively
(Lp+Lg+Lr=stride
length).

Variation in velocity during one paddling cycle. The plot was obtained from
one video sequence (Bird 5) and smoothed by a stepwise 5-point moving average
for easy viewing. The plot shows the kinematic parameters defined by Weihs
(1974) for the burst-and-glide
model (L1, L2, Ui,
Uf), and the additional parameters defined in the present
study (Lp, Lg, Lr,
Ur, tp, tg,
tr); see List of symbols for the description of each
parameter. The bottom bar indicates the division of the paddling cycle into
three stages based on the motion of the feet relative to the body (stroke,
glide and recovery). Average swimming speed for this cycle was 1.72 m
s–1. Note that the bird did not swim at a constant speed but
rather accelerated during the power stage and decelerated during the glide and
recovery stages in a burst-and-glide swimming pattern.

Stride length, paddling frequency and average swimming speed were measured
from all the sequences available (95 sequences from 10 birds). However, for
the calculation of the drag coefficients we used only the best 35 sequences
(from 10 birds) that had the same speed at the start and at the end of the
cycle and the smoothest velocity profiles, with distinct acceleration and
deceleration. In a few cases, data smoothing, using 3 or 5 point stepwise
moving averages, was necessary to determine the exact point of transition
between tp and tg, on the velocity
profiles. The sequences used were the ones where the birds were swimming as
straight as possible (no sideslips and minimal vertical deviations of the
center of mass).

Estimation of drag

The instantaneous speed profiles from each of the 10 birds were used to
evaluate the drag coefficients of the birds, both when actively swimming and
during gliding. We assumed that during the glide and recovery stages, the body
moves due to inertia, and thrust is not added to the system so that the
Cd of the glide and recovery can be calculated directly
from the deceleration of the birds. This assumption is trivial for the glide
stage, but in the recovery stage, the forward motion of the feet may
presumably add some forward thrust (from inertia). Here we assume that this
thrust is negligible since it is likely cancelled by the increased drag on the
feet, as they move against the swimming direction. The velocity profiles
indicate that the birds indeed decelerated during the recovery stage
(Fig. 2).

For the calculation of the drag coefficient during the glide stage we used
the technique suggested by Bilo and Nachtigall
(1980), based on the average
linear slope (C) of the inverse instantaneous speed
(U–1) vs time:
(4)
where C is the slope of the least-square linear regression between
U–1 and time (obtained from the glide data),
m is the virtual mass of the birds (bird mass + added mass;
Stelle et al., 2000),
A is taken as volume of the bird to the 2/3 power, and ρ is water
density (103 kg m–3). Virtual mass (m) is
calculated from the mass of the bird (M) in kg, the added mass
coefficient calculated for the bird in the mean angle-of-attack during the
glide (Ca,, see morphometric measurements) and
the volume of the bird (V) in m3. It is thus calculated
as:
(5)
The calculations of Cd were made once for the glide stage
(Cd,glide) and then for the combined glide + recovery
stage (Cd,glide+recovery). After
Cd,glide and Cd,glide+recovery are
calculated, the Cd for the recovery stage
(Cd,recovery) can be isolated either directly from the
kinematic data of the deceleration during the recovery, or arithmetically,
treating the Cd,glide+recovery as the mean contribution of
the Cd,glide and the Cd,recovery combined:
(6)
where Cd,glide+recovery and Cd,glide
were calculated from Eqn 4 above, and the distances Lg and
Lr are taken from the kinematic data.

We can now use the burst-and-glide model to calculate
Cd for the burst (power) stage
(Cd,stroke). The burst-and-glide model is based on a
two-stage swimming behavior. It defines L1 as the distance
traveled during the acceleration (burst) and L2 as the
distance traveled during the deceleration and provides equations to calculate
each of them from Ui and Uf (see
Appendix). We used the fact that L1 and
L2 can be measured directly from the kinematic data to
solve these equations for α in the burst-and-glide model. The factorα
is the ratio between the drag coefficient during the burst stage and
the drag coefficient during the glide. We replaced L2 in
the original calculation with our
Lg+Lr and used Lp
instead of the original L1. The modified equations are
therefore:
(7)
where
(8)
where
Ūi=Ui/Ue and
Ūf=Uf/Ue;
Ue is a reference velocity, chosen, for convenience, to be
larger than the highest measured velocity so that the normalized values are
always <1. Weihs (1974)
called this speed the maximum achievable sustained swimming speed.

The factor α may now be isolated from Eqn 7 and solved using the
values of Lp, Lg,
Lr, Ui, Uf and
Ue. Except for the latter, all parameters are available
from the kinematic data. For Ue we used the value 2.5 m
s–1, which was slightly higher than the maximum
Uf observed in our data (2.41 m s–1).
Once α is obtained, the Cd of the power stage
(Cd,stroke) can be obtained from the definition ofα
:
(9)
where Cd,glide+recovery was obtained from Eqn 4.

We now have the drag coefficient for each of the stages (stroke, glide and
recovery). To calculate the instantaneous drag force (at
Re>105) from the drag coefficients at any point in time
along the cormorant paddling cycle, one needs to substitute the appropriate
Cd and instantaneous speed (U) into Eqn 1 and
isolate D as:
(10)
The average drag for the entire paddling cycle is the integral of Eqn 10 over
time, divided by the paddling cycle duration:
(11)
where tp, tg and
tr are the durations of the power glide and recovery
stages, respectively (Fig.
2).

The burst-and-glide model was developed to evaluate the energetic advantage
of using burst-and-glide. If we continue to solve the original model we can
obtain the energetic advantage (R) of combining glides in the
paddling cycle, as opposed to constant paddling at constant swimming speed,
from the following equations (modified from
Videler and Weihs, 1982; see
Appendix):
(12)(13)R is the ratio of the energy expended during burst-and-glide
vs the energy expended for swimming constantly at the same average
speed. In the case where there is no benefit in gliding R=1, and as
R becomes smaller, the energetic advantage in burst-and-glide, as
opposed to swimming at constant speed, will be greater. However, in our
specific case, simply solving for R will result in an exaggerated
value. This is because we combined the recovery stage with the glide for the
drag estimation. The α value that we calculated above (as a means to
find Cd,stroke) considers only the stroke stage as `active
swimming', while in practice, the recovery stage is part of the active
swimming as well. To find α that is relevant for the calculation of the
energetic advantage in combining glides into the paddling cycles that contain
only stroke and recovery, we repeated the analysis, this time with the
recovery included in the burst stage of the burst-and-glide model.

We defined a new drag coefficient for the stroke + recovery stages
(Cd,propulsion) and recalculated α (Eqn 7), this
time replacing L1 with
(Lp+Lr), and Ui
with Ur (the velocity at the beginning of the recovery
stage). The above mentioned Eqn 7 and 8 then become:
(14)
and
(15)
where
Ūr̄=Ur/Ue.

The new α obtained is defined as:
(16)
We then continue to use the burst-and-glide model, replacing
L1 in the original model with
(Lp+Lr), and Ui
with Ur. Thus, Eqn 12 and 13 above become:
(17)
and
(18)
This Rp is the energetic saving, adjusted for the special
case of three-stage paddling. It gives the ratio of the energy required for a
paddling cycle including a glide vs the energy needed to swim
constantly at the same average swimming speed, with a paddling cycle that
includes only stroke and recovery.

Data analysis

Thirty five sequences (from ten different birds) were fully analyzed for
the drag coefficients and burst-and-glide parameters (R,
Rp) while a larger sample size was used to analyze gait,
paddling frequency and stride length (95 sequences). Statistical significance
for correlations between variables was tested using Statistica (StatSoft,
Tulsa, OK, USA). To ensure no bias from pseudo-replications or the unequal
number of trials per bird, statistical significance of all correlations that
are reported here for all the trials were tested a second time using mean
values per bird (N=10 birds). Similarly, the mean values reported
throughout this work are the means of all birds (N=10), calculated by
averaging all values from sequences from a single bird and then averaging for
all birds. Values of statistical variation quoted in the text and figures are
thus standard error of mean (s.e.m.) where
N=10 birds. Tables 2
and 3 are an exception. These
tables present the means of each bird calculated from small and unequal sample
size. We therefore chose to present variance in these tables as the standard
deviation (s.d.).

The calculated drag coefficient for each stage of the paddling cycle of
the great cormorant and the resulting average drag per cycle

Results

The morphometric measurements performed on carcasses are summarized in
Table 1.

The swimming speed of the cormorants in the video sequences was
1.51±0.058 m s–1 (N=10 birds). Mean swimming
speed calculated for each bird in the experiment
(Fig. 3) was correlated with
the bird's body mass (r=0.75, P<0.015, N=10) and
body length (r=0.87, P<0.015, N=10). The
paddling frequency was positively correlated with swimming speed
(r=0.56, P<0.001, N=95 sequences). This was
achieved mainly by shortening the duration of the glide stage as the paddling
frequency increased (r=–0.86, P<0.001,
N=95 sequences) while the duration of the stroke and recovery stages
remained unaffected (stroke: r=0.009, P>0.93; recovery:
r=0.035, P>0.74) (Fig.
4A). Taking this into account, the relation between the duration
of the glide stage and the paddling frequency suggests a maximal paddling
frequency of approximately 2.7 Hz when glide duration is zero
(Fig. 4A,B).

Table 2 summarizes the
kinematic data used for the drag estimation and
Table 3 summarizes the various
drag coefficients, R and Rp, calculated from the
model. The Re for each bird, based on body length and average speed
(Table 3), ranged between
0.9×106 and 1.4×106 (mean=
1.12×106±5.92×104). The average
Cd,glide calculated for each bird, using
V2/3 as a reference area, was 0.143±0.011
(N=10 birds), but ranged between a minimum of 0.091 to a maximum of
0.180. The Cd,glide calculated for each sequence was
negatively correlated with the Re and mean swimming speed calculated
for the same sequence (r<–0.62, P<0.001,
N=35 in both cases). When the average Cd,glide
for each bird was compared with the average Re (or the average speed)
for the same bird (giving equal weight to all birds) the correlation improved
(r<–0.93, P<0.001, N=10). Each
Cd,glide obtained from a video sequence was positively
correlated with the average angle-of-attack of the body during the glide in
that sequence (r=0.50, P<0.003, N=35). A similar
relation between the average angle-of-attack during the stroke and
Cd,stroke was not significant (r=–0.11,
P>0.54, N=35 sequences). The average angle-of-attack
during the glide was negatively correlated to the average swimming speed
(r=–0.65, P<0.001, N=35), so while
swimming faster the birds were swimming at smaller angle-of-attack and
vice versa. The average angle-of-attack during the stroke stage was
not significantly correlated to the average swimming speed
(r=–0.29, P>0.09, N=35) or to the average
swimming speed during the stroke phase (r=–0.21,
P>0.22, N=35). Cd,stroke was
positively correlated with the average swimming speed (r=0.41,
P<0.015, N=35).

The average swimming speed observed for each cormorant in the experiment
relative to body length. Each point is the mean swimming speed obtained from
5–10 sequences taken of the same bird (values are ±
s.e.m.).

The average Cd,recovery and the
Cd,stroke were 0.273±0.054 and 0.432±0.040
respectively (N=10 birds). The average
Cd,recovery and Cd,stroke for each
bird were 2.6±0.71- and 3.2±0.57-fold higher, respectively, than
Cd,glide of the same bird (N=10). The average
drag for the entire paddling cycle (as calculated from Eqn 11) for each bird
was 5.5±0.68 N (N=10). The drag was highest during the stroke,
then decreased considerably during the glide, and increased slightly during
the recovery stage (Fig. 5).
The burst-and-glide model predicts an average Rp value of
0.99±0.077 and a R value of 0.68±0.078
(Table 3). Thus, when the
recovery stage is considered as part of the glide, the average energetic gain
of the birds from the non-constant swimming speed is 32%. However, if the
recovery is considered as part of the burst stage, then on average there is no
energetic saving from using burst-and-glide. The values of R and
Rp were negatively correlated with the average swimming
speed (r<–0.68; P<0.001; N=35 for both
R and Rp), suggesting that the faster birds were
gaining more from burst-and-glide while the slower birds were actually losing
from it relative to an idealized constant swimming
(Fig. 6A). The α
calculated from the burst-and-glide model (using Eqn 7 or 14) increased
non-linearly with the increase in swimming speed
(Fig. 6B). At low speeds,α
was close to 1 and then increased sharply up to 8 in the fastest
birds.

The duration of each stage in the paddling cycle of the great cormorant (A)
and the proportion of the power (burst) and deceleration (glide + recovery)
stages in the paddling cycle (B) in relation to paddling frequency. Note that
the glide duration decreased with the increase in paddling frequency. At a
paddling frequency of 2.7 Hz, the duration of the glide phase was diminished
to almost zero so that the power and recovery stages each occupy ∼50% of
the paddling cycle duration.

Discussion

The present study shows that in order to swim faster, cormorants shorten
their paddling cycle by shortening the duration of the glide stage within each
cycle. Clark and Bemis (1979)
reported a similar trend in penguins and suggested that this either allows the
propelling muscles to operate at maximum efficiency over a range of swimming
speeds or allows an energy saving from burst-and-glide swimming. In the case
of the cormorant, an additional advantage is derived from the fact that the
recovery stage occurs when the speed of the body is minimal. Thus, the
increase in the drag coefficient during the recovery stage is compensated for
by the lower swimming speed at the end of the paddling cycle, maintaining a
relatively low drag (Fig.
5).

During the glide, the body is at the smallest angle-of-attack, the neck is
aligned with the swimming direction, and the feet are trailing behind the
body. At this posture, the bird can optimally use its momentum to keep moving
at relatively low drag. During propulsion (stroke and recovery), the tilt of
the body is increased and the feet are moved from their posterior position
under the tail. This results in an increase in the drag coefficient, and thus
the drag of active swimming is higher than during gliding. The duration of the
paddling cycle is determined by the length of the glide stage. When paddling
frequency increases, propulsion occupies a more significant portion of the
paddling cycle (Fig. 4B) and
this affects the drag of the bird. Therefore, the ability to calculate a drag
coefficient for each stage of the swimming cycles, using simple kinematics,
allows the more accurate calculation of the drag during swimming for a wide
range of paddling cycles that combine glides and propulsion.

While the drag coefficient for the glide stage was lower than the drag
coefficient for the recovery and stroke stages, its value was affected by the
need to counter buoyancy. Cormorants are positively buoyant and use
hydrodynamic lift to counter their tendency to float while swimming underwater
(Ribak et al., 2004). The
magnitude of the lift force is dependent on swimming speed and on the
angle-of-attack of the body. We show here that the birds that were swimming
slower had higher drag coefficients for the glide stage. This is because the
slower birds tilted their body at a larger angle-of-attack to compensate for
the reduced speed and achieve sufficient lift to counter their buoyancy. This
tilt increased their drag coefficient, as evident from the significant
correlation between Cd,glide and the tilt of the body.
Thus, the actual drag encountered by cormorants swimming at shallow depths is
greatly affected by the need to counter buoyancy. Buoyancy is considered the
primary mechanical constraint of diving water birds, but this constraint is
depth-dependent (Wilson et al.,
1992; Lovvorn and Jones,
1991). Here we show that although the birds have managed to cope
with the mechanical constraint of buoyancy during shallow submerged swimming,
there exists an energetic trade-off in which the birds are required either to
swim faster or swim at a larger angle-of-attack, increasing the drag in either
case and thus making shallow foraging more costly.

The distribution of the momentary drag along an average paddling cycle of
the great cormorant. Momentary drag is calculated using the
Cd average for each of the paddling stages. To average all
the analyzed sequences despite the differences in cycle duration, the
x-axis is the proportion of the cycle duration that is normalized by
dividing the period of each stage (stroke, glide and recovery) by the mean
duration of that stage. To allocate equal weight to all birds, the sequences
of each bird were first averaged and then means ±
s.e.m. of all birds were calculated.

Unlike in the glide stage the Cd,stroke was not
statistically correlated with the average angle-of-attack of the body during
the stroke. This may be due to the fact that the angle-of-attack changes
throughout the stroke stage while our Cd,stroke is an
average for the entire stage. Another explanation is that during the stroke
the tail changes orientation and the feet move beneath the body, thus changes
in the drag coefficient during propulsion are not limited to the increase in
the angle-of-attack. Cd,stroke was, however, positively
correlated with the average swimming speed. Thus as swimming speed increased,
Cd,glide decreased while Cd,stroke
increased, and this resulted in the increase in α observed here
(Fig. 6B).

The birds in our study differed only slightly in body length
(0.71–0.82 m), and to a greater extent in average swimming speed
(1.3–1.75 m s–1). Hence the variation in Re
between birds was mainly due to the difference in swimming speeds. Re
was negatively correlated with Cd,glide. Such a relation
may be due to a reduction in the thickness of the boundary layer
(Hoerner, 1965). In our case,
however, it is more likely that the decrease in Cd,glide
with speed is due to the above-mentioned need to tilt the body further during
slow swimming. Regardless of the precise reason for the decrease in
Cd,glide at higher Re, this relationship explains
why each of our birds had a fairly distinct swimming speed
(Fig. 3) that was correlated
with body size. The average thrust for a complete cycle, developed by each
bird, should equal its drag. If we assume that the thrust developed by each
bird during the paddling is proportional to the bird's size (i.e.
thrust/A=constant), then the birds should glide at similar values of
Cd,glideU2. Multiplying the square of the
average swimming speed and the average Cd,glide obtained
for each bird yields a constant value (0.309±0.0089; N=10)
that was not significantly correlated with the length of the birds or with
their body mass (r<0.24, P>0.5, N=10 for both
cases). Thus, the observed variation in swimming speeds among birds was a
consequence of the birds' drag coefficients during the glide.

The `burst-and-glide' parameters R or Rp (A)
and α (B) in relation to the average swimming speed of the great
cormorant in each paddling cycle. Open symbols denote calculated values when
the recovery is treated as part of the burst stage and full symbols are
calculated values when the recovery is treated as part of the glide stage (see
text). Solid horizontal lines mark the proportion of 1, where the energetic
cost of burst-and-glide is exactly the same as for swimming at constant speed.
Broken vertical lines in A mark the threshold average speed where the least
square regression of the points (broken line for Rp, solid
line for R) intersects with the value of 1. At this swimming speed
the energetic benefit from burst-and-glide changes from gain (R<1)
into loss (R>1). In B, values ≤1 imply that the drag
coefficient for passive drag is as high as or higher than the coefficient for
active swimming. Note that the open symbols represent a more conservative
approach for estimating the energetic advantage from burst-and-glide.

The average Cd,glide reported here is intermediate
between the values calculated for a model and for carcasses of the Brandt's
cormorant (P. pencillatus) towed in a water tunnel by Lovvorn et al.
(2001). Our value for
Cd,glide at Re=1.12×106 using
surface area as a reference (see Table
1 for the conversion factor) was 0.0186±0.0014. This value
is higher than that reported for a featherless model, but slightly lower than
the value reported for a frozen carcass (∼0.008 and ∼0.022,
respectively, at Re=106; data taken from figs 6 and 7 in
Lovvorn et al., 2001). Drag
measurements on carcasses of birds tend to overestimate the drag coefficient
due to feather fluttering (Pennycuick et
al., 1988; Tucker,
1990). The featherless model, on the other hand, underestimates
the true drag of the birds during the glide by measuring drag at zero lift
(θ=0).

Our original application of the burst-and-glide model was to calculate drag
coefficients and drag for the entire paddling cycle of the cormorants.
However, the model also provides a tool with which to explore the energetic
benefit (if existing) of the paddling cycle of these birds. The values of
Rp and R that were calculated in our case are
rather hypothetical because the burst-and-glide model was designed for animals
such as fish that can develop constant thrust. In the original model, the
energy required for a fish to overcome water resistance during swimming is
compared between swimming at constant speed and swimming at periodically
varying speeds oscillating around the same average swimming speed. The
paddling of the cormorant, with both feet moving together, is different from
this scenario due to existence of the recovery stage. Cormorants can never
truly swim at constant speed since the body decelerates during the recovery
stage. Therefore, the R and Rp we obtained from
the model is the energetic saving from the burst-and-glide swimming pattern
relative to the energy required for swimming at constant speed, which is only
theoretical. Another difference is the fact that although the recovery is a
continuation of the glide (since the body continues to decelerate), it is also
part of the burst stage, since without it there is no constant paddling. To
deal with these special characteristics of paddling we included the recovery
stage in the burst, and therefore compared the energetic saving from combining
glides with propulsion, as opposed to paddling constantly at the same average
speed.

The values we found for Rp
(Table 3,
Fig. 6A) suggested that most of
our birds were gaining little from combining glides with the paddling cycle.
As the average swimming speed became increasingly lower than
Ue, the energetic advantage from burst-and-glide
diminished to a point where, according to the model, the birds that would
propel constantly would save more energy than the birds that added glides to
their paddling cycle (Rp>1). The inverse relation
between swimming speed and Rp is predicted by the
burst-and-glide model. Weihs
(1974) has shown that for a
given α, R increases (energy saving decreases) when the average
swimming speed is reduced with respect to Ue. Therefore,
for each value of α there is a minimum average swimming speed where
burst-and-glide becomes inefficient. Swimming below this threshold speed will
result in R>1. In addition to this the threshold speed decreases
as α becomes higher. This way, birds with a higher α benefit from
burst-and-glide even at slower swimming speeds. The slower birds in our
experiments had the lowest values of α due to the fact that their
Cd,glide was relatively higher. This elevated
Cd,glide was due to the need to counter buoyancy at
shallow depths. It follows that at deeper depths, where buoyancy of the birds
is reduced, Cd,glide will decrease (due to a decrease in
the angle-of-attack). Cd,stroke does not seem to be
related to buoyancy as Cd,glide, and if this is indeed the
case then α should increase with depth, making burst-and-glide more
beneficial. This conjecture can be validated by Eqn 7 and 14 above. If we
assume that birds swim with the same Ui and
Uf but at deeper depth so that their buoyancy is reduced,
and also Cd,glide, then the birds deceleration during the
glide stage should reduce as well. Since Ui and
Uf remain the same, the birds glide for longer to
decelerate from Uf to Ui (or from
Uf to Ur), thereby traversing a larger
distance of Lg. Since Lg is the
numerator in Eqn 7 and 14, the larger Lg for the same set
of Ui and Uf results in a largerα
. Therefore, the ineffective burst-and-glide observed here
(Rp≥1) may prove beneficial at larger depths where
buoyancy of the birds is lower.

Fig. 6 shows that the
transition from benefit to loss (Rp=1) from adding glides
to the paddling cycle occurs at an average swimming speed of about 1.5 m
s–1, which is identical to the mean swimming speed observed
in our birds. Therefore, only birds that swim at high speeds may gain from
adding a glide to their paddling cycle, relative to constant paddling. Yet,
all the birds were adding glides to all paddling cycles, regardless of their
average speed. The reason is that Rp is a conservative
estimate. As mentioned above, the recovery stage is part of the propulsion, on
one hand, but it is also part of the deceleration, on the other. When we add
the recovery to the stroke stage, we reduce the average acceleration during
the burst stage, lower the average drag coefficient for the burst stage
(Cd,propulsion), lengthen the burst stage, and shorten the
deceleration stage. This makes paddling less costly relative to gliding
(α becomes smaller) and hence the gain in gliding diminishes. However,
during the recovery stage the bird continues to move forward
(Lr) at a lower drag coefficient and without adding
thrust. If the model is rearranged, with the recovery stage added to the glide
stage, then the burst stage shortens, it has a higher drag coefficient and the
deceleration stage is longer. The R calculated this way (Eqn
7–13) is mostly below 1, and the threshold swimming speed shifts to∼
1.0 m s–1 (Fig.
6). This speed is similar to the minimal average swimming speed
observed from our birds (1.154 m s–1; bird 8). The true
energetic benefit and threshold swimming speed are somewhere in between these
extreme values for R and U.

The R and Rp values express the energy savings
from using burst-and-glide vs swimming at constant speed. A
hypothetical scenario where cormorants are able to swim at constant speed is
one where the birds will use alternate paddling underwater. Cormorants only
use alternate paddling while swimming on the surface and use synchronized
strokes only underwater (Ancel et al.,
2000; Lovvorn,
1991). It is thus interesting to note that the transition between
alternate paddling and synchronized strokes should occur at a swimming speed
of 1.0–1.5 m s–1, which is similar to the findings that
surface-swimming Brandt's cormorants switch from surface swimming to submerged
swimming at a speeds of about 1.3 m s–1
(Ancel et al., 2000). The birds
dive primarily to avoid the higher work against drag at the surface, but the
point of transit from surface swimming to submerged swimming also encompasses
a change in gait, from alternate paddling to synchronized strokes, and this
point of transit fits well with the energy saving threshold found in the
present study. However, further research is needed to test the suggestion that
in paddlers the shift from alternate to synchronized paddling occurs in
agreement with burst-and-glide predictions. If the cormorant has indeed a
higher α when swimming at the same swimming speed but at deeper depths,
then the associated shift of the threshold swimming speed to slower speeds (as
predicted by the burst-and-glide model;
Weihs, 1974;
Fig. 6A) will ensure that
burst-and-glide will have a higher benefit with depth making alternate
paddling underwater inefficient in practical swimming speeds. The closely
related anhinga, however, is similar to the cormorant in shape but has a
different foraging strategy. Due to unusual plumage wettability it relies on
hydrostatic mechanisms to become neutrally buoyant underwater
(Johansgard, 1993). This
allows it to swim very slowly underwater and stalk pray (unlike the active
pray chasing in the cormorant). Alternate paddling underwater as well as
synchronized strokes have been reported for anhingas
(Owre, 1967), and this raises
the question whether this behavior is associated with burst-and-glide
efficiency, as discussed above. If the bird is neutrally buoyant its α
should remain constant with depth and so should the trade-off speed. Therefore
during slow swimming, anhingas may profit from shifting to alternate paddling
when burst-and-glide is no longer effective. Unfortunately this suggestion
cannot be evaluated at present due to lack of observations on the relationship
between paddling gait and horizontal swimming speed at various depths in
anhingas and cormorants.

List of symbols

a

acceleration

A

area (general)

α

ratio between the Cd for active swimming and
Cd for gliding

Aw

surface area

C

average linear slope of the inverse instantaneous swimming speed
vs time

c

components of drag that are not speed dependant
(0.5ρACd)

Ca

added mass coefficient (general)

Ca(θ)

added mass coefficient when the body is tilted at an angle of
attack=θ

Cax

added mass coefficient for motion parallel to the length of a
body

Cay

added mass coefficient for motion perpendicular (vertical) to the
length of a body

Cd

drag coefficient (general)

Cd,glide+recovery

average drag coefficient during the glide + recovery stage

C

d,glide drag coefficient during the glide stage

Cd,propulsion

average drag coefficient during the power + recovery stage

C

d,recovery drag coefficient during the recovery
stage

C

d,stroke drag coefficient during the power (acceleration)
stage

D

hydrodynamic drag force

ρ

density of water (103 kg m–3)

F

force (general)

l

body length

L1

distance traversed during the burst in the original burst-and-glide
model

L2

distance traversed during the glide in the original burst-and-glide
model

Lg

distance traversed during the glide stage

Lp

distance traversed during the power (acceleration) stage

Lr

distance traversed during the recovery stage

M

mass

m

virtual mass (mass of the body + added mass of water)

R

relative energetic saving from burst-and-glide (in the case of the
cormorant: when the recovery stage is considered as part of the
glide)

Re

Reynolds number

Rp

relative energetic saving from burst-and-glide when the recovery stage
is considered as part of the burst

tg

duration of the glide stage

tp

duration of the power stage

tr

duration of the recovery stage

θ

angle-of-attack

U

speed (general)

Ūc

average swimming speed in the burst-and-glide model divided by
Ue

Uc

average swimming speed in the burst-and-glide model

Ue

maximum achievable swimming speed

Ūf

instantaneous speed at the end of the burst stage divided by
U

Uf

instantaneous speed at the end of the burst stage

Ūi

instantaneous speed at the start of the burst stage divided by
Ue

Ui

instantaneous speed at the start of the burst stage

Ūr

instantaneous speed at the start of the recovery stage divided by
Ue

Ur

instantaneous speed at the start of the recovery stage

V

volume

V2/3

volume of the bird to the 2/3 power (representing area)

y

half the body height measurement at a point along the body's
length

υ

kinematic viscosity of fresh water (at
20°C=∼10–6 m2
s–1)

z

half the body width measurement at a point along the body's
length

Appendix

Burst-and-glide is an energy saving locomotion strategy where, instead of
swimming at a constant speed, an animal accelerates from an initial velocity
Ui to a final velocity Uf during a
short stage termed `burst'. It then glides passively (without producing
thrust), decelerating until it reaches the initial velocity
Ui (glide stage). During this cycle the animal travels at
an average swimming speed Uc. To evaluate when
burst-and-glide is energetically efficient, the energy expended from this
swimming strategy is compared to the energy expended during swimming at
constant speed Uc. This is done by calculating the energy
ratio R, which is the ratio between the energy needed to
burst-and-glide vs the energy needed to swim constantly, where in
both cases the average swimming speed is Uc. Weihs
(1974) showed that R
can be calculated from Ui and Uf,
(which are obtained directly from observations) if two additional parameters,
Ue and α, are known. Ue is the
maximum sustained velocity achievable during maximum thrust. It is used to
normalize all speeds to a value ≤1. The parameter α is the factor
describing the increment in drag when an animal perform swimming motions as
oppose to gliding passively (rigid body). In his analysis, Weihs
(1974) developed equations for
the distances traversed (L1 and L2)
and the time period (t1 and t2) during
the burst and the glide stages, respectively, and for Uc.
In final form these equations are listed below.

The distance traveled during the burst stage (eqn 21 in
Weihs, 1974):
(A1)

The distance traveled during the glide stage (eqn 24a in
Weihs, 1974):
(A2)

The average swimming speed (eqn 25 in
Weihs, 1974)
and
(A5)m is the virtual mass (mass + added mass of water accelerated with
the body) and c is the component of drag that is not dependant on
speed (from the drag term
D=0.5ρACdU2=cU2r̄c=0.5ρACd).
Both are unknown but will not be required for the final calculation of
R, as explained below.

This solution has an advantage in the calculation since using the ratio
L2/L1 both m and c
cancel out and the final equation can be calculated using only
Ui, Uf, Ue andα
. Thus:
(A8)R is the ratio between the energy required for burst-and-glide and
the energy required for constant swimming at Uc. For
values of R<1, burst-and-glide saves energy relative to constant
swimming and when R>1, constant swimming is more energetically
efficient.

ACKNOWLEDGEMENTS

This research was funded by the Israeli Ministry of Science and Technology
under contract no. 3920. We are grateful for the assistance of Ohad Agranat,
Ruth Almon, Shani Set and Shoshana Goldenberg in the experiments. Comments
from two anonymous reviewers greatly improved this manuscript. The Video
Department of the Technion provided technical support.

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